Mathematical Sciences: Integrable Models in Mathematics and Physics

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This research involves the quantum field theory of fermions on the Poincare disk and the hard hexagon model - a lattice gas model in statistical mechanics. The research on the quantum theory of fermions will involve expressing the Schwinger functions in terms of certain nonlinear deformation equations. These equations will then be analyzed both in the Euclidean limit and in the conformal field theory limit. An important aspect of this work is the extension of the existing theory to the noneuclidean setting which may lay the groundwork for extensions to Riemann surfaces of a more general type. The research on the hard hexagon model will involve derivations of differential equations for the partition function and density functions. An attempt will be made to extend these equations to the hard square lattice gas, a model that is 'not solvable' by present techniques. This research is in the general area of theoretical physics and deals with isomonodromy deformations and applications to conformal field theory, statistical mechanics, and scattering theory.